Integer Partitions Research Articles

FAMILY OF CONGRUENCES FOR (2, β) REGULAR BIPARTITION TRIPLES

Though congruences have their limitations, they have significant impor- tance in the field of number theory and helps in proving many interesting results. Thus, this article has adopted the technique and properties of congruences to iden- tify and prove a set of congruent properties for integer partition. The partition of a positive integer is a way of expressing the number as a sum of positive in- tegers. One such partitions known as regular bipartition triple are discussed in this article. New congruences modulo even integers and modulo prime (p 5) powers are derived for (2, β) regular bipartition triples. Also infinite families of congruences modulo 2 for some (2, β) regular bipartition triples are derived. The theorems stated in this article are proved using the q series notation and some of the prominent results such as Euler’s pentagonal number theorem and Jacobi’s triple product identities. There are certain lemmas which are derived using these results that help in proving the major results of this article.

On the Structural Properties and Some Topological Indices of Young-Fibonacci Graphs

In this paper, we study Young Fibonacci graphs Gn, a special family of graphs that are constructed with the help of integer partitions. Young diagrams are also used in the construction of graphs. The family of graphs is rich in structure. Thus, we investigate various properties of the family of graphs which include degree based structure and topological in-dices. Topological indices like Zagreb Index, Wiener Index, Randic Index and Connective Eccentricity Index of these graphs are computed. We also study the eigenvalues and energy of the graph.

Partition identities related to the minimal excludant

In this work, we study the newer statistic “minimal excludant” for integer partitions, and obtain several interesting partition identities. In particular, we find a new and simple refinement of Euler's celebrated partition identity. As an unexpected byproduct of this work, we establish a bivariate form of Ramanujan's special theta function φ(q).

-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION

AbstractA partition of a positive integer n is called $\ell $ -regular if none of its parts is divisible by $\ell $ . Denote by $b_{\ell }(n)$ the number of $\ell $ -regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$ .

Enumerating partitions arising in homotopy theory

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.

Double generalized majorization and diagrammatics

In this paper we show that the generalized majorization of partitions of integers has a surprising completing-squares property. Together with the previously obtained transitivity-like property, this enables a compelling diagrammatical interpretation.Apart from purely combinatorial interest, the main result has applications in matrix completion problems, and representation theory of quivers.

ON THE NUMBER OF NEARLY SELF-CONJUGATE PARTITIONS

AbstractA partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

Alternating sign property of the perfect matching derangement graph

It was conjectured in the monograph [9] by Godsil and Meagher and in the article [10] by Lindzey that the perfect matching derangement graph M2n possesses the alternating sign property, that is, for any integer partition λ=(λ1,…,λr)⊢n, the sign of the eigenvalue ηλ of M2n is given by sign(ηλ)=(−1)n−λ1. In this paper, we prove that the conjecture is true. Our approach yields a recurrence formula for the eigenvalues of the perfect matching derangement graph as well as a new recurrence formula for the eigenvalues of the permutation derangement graph.

A Measure-on-Graph-Valued Diffusion: A Particle System with Collisions and Its Applications

A diffusion-taking value in probability-measures on a graph with vertex set V, ∑i∈Vxiδi is studied. The masses on each vertex satisfy the stochastic differential equation of the form dxi=∑j∈N(i)xixjdBij on the simplex, where {Bij} are independent standard Brownian motions with skew symmetry, and N(i) is the neighbour of the vertex i. A dual Markov chain on integer partitions to the Markov semigroup associated with the diffusion is used to show that the support of an extremal stationary state of the adjoint semigroup is an independent set of the graph. We also investigate the diffusion with a linear drift, which gives a killing of the dual Markov chain on a finite integer lattice. The Markov chain is used to study the unique stationary state of the diffusion, which generalizes the Dirichlet distribution. Two applications of the diffusions are discussed: analysis of an algorithm to find an independent set of a graph, and a Bayesian graph selection based on computation of probability of a sample by using coupling from the past.

Permutation graphs and the weak Bruhat order

Permutation graphs are graphs associated with permutations where edges represent inversions. We study different classes of permutation graphs and  isomorphic permutation graphs. A complete answer of a basic isomorphism question is given for trees. A connection to the majorization order for integer partitions is established, in connection with the notion of inversion vectors. Finally, we study special transitivity properties related to inversions of permutations.

A Proof of the $$\frac{n!}{2}$$ Conjecture for Hook Shapes

A well-known representation-theoretic model for the transformed Macdonald polynomial \({\widetilde{H}}_\mu (Z;t,q)\), where \(\mu \) is an integer partition, is given by the Garsia–Haiman module \({\mathcal {H}}_\mu \). We study the \(\frac{n!}{k}\) conjecture of Bergeron and Garsia, which concerns the behavior of certain k-tuples of Garsia–Haiman modules under intersection. In the special case that \(\mu \) has hook shape, we use a basis for \({\mathcal {H}}_\mu \) due to Adin, Remmel, and Roichman to resolve the \(\frac{n!}{2}\) conjecture by constructing an explicit basis for the intersection of two Garsia–Haiman modules.

Semi-modular forms from Fibonacci–Eisenstein series

In a 2021 paper, M. Just and the second author defined a class of “semi-modular forms” on $${\mathbb {C}}\backslash {\mathbb {R}}$$ , in analogy with classical modular forms, that are “half modular” in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.

Asymptotics for the twisted eta-product and applications to sign changes in partitions

We prove asymptotic formulas for the complex coefficients of (zeta q;q)_infty ^{-1}, where zeta is a root of unity, and apply our results to determine secondary terms in the asymptotics for p(a, b, n), the number of integer partitions of n with number of parts congruent a modulo b. Our results imply that, as n rightarrow infty , the difference p(a_1,b,n)-p(a_2,b,n) for a_1 ne a_2 oscillates like a cosine when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of {p(a,b,n)}_{1 le a le b}.

Divisibility and distribution of mex-related integer partitions of Andrews and Newman

Andrews and Newman introduced the minimal excludant or “[Formula: see text]” function for an integer partition [Formula: see text] of a positive integer [Formula: see text], [Formula: see text], as the smallest positive integer that is not a part of [Formula: see text]. They defined [Formula: see text] to be the sum of [Formula: see text] taken over all partitions [Formula: see text] of [Formula: see text]. We prove infinite families of congruence and multiplicative formulas for [Formula: see text]. By restricting to the part of [Formula: see text], Andrews and Newman also introduced [Formula: see text] to be the smallest odd integer that is not a part of [Formula: see text] and [Formula: see text] to be the sum of [Formula: see text] taken over all partitions [Formula: see text] of [Formula: see text]. In this paper, we show that for any sufficiently large [Formula: see text], the number of all positive integer [Formula: see text] such that [Formula: see text] is an even (or odd) number is at least [Formula: see text].

Partitions, their classes, and multicolour evenness

We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe . Interactions measure the average evenness of colour sets (partitioning sets) of . For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.

Mixed-Up-Ness or Entropy?

Simple SummaryThe second law of thermodynamics has a mystical appeal in disciplines with tenuous connections to its origins. We hypothesize that many of these appeals instead should be to another principle heretofore unrecognized: the law of mixed-up-ness (LOM). Instead of using a number such as entropy to characterize randomness, non-thermodynamic systems can be arranged in simple diagrams according to their degree of mixed-up-ness. Curiously, the evolution of such systems from degrees of low to high mixed-up-ness is both consistent with and richer than the principle of increasing entropy.Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect “order” to states of total “disorder”. Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.

Quantum black holes, partition of integers and self-similarity

In this paper, we take the view that the area of a black hole’s event horizon is quantized, [Formula: see text], and the associated degrees of freedom are finite in number and of fermionic nature. We then investigate general aspects of the entropy, [Formula: see text], our main focus being black hole self-similarity. We first find a two-to-one map between the black hole’s configurations and the ordered partitions of the integer [Formula: see text]. Hence, we construct from there a composition law between the subparts making the whole configuration space. This gives meaning to black hole self-similarity, entirely within a single description, as a phenomenon stemming from the well-known self-similarity of the ordered partitions of [Formula: see text]. Finally, we compare the above to the well-known results on the subleading (quantum) corrections, which necessarily require different (quantum) statistical weights for the various configurations.

Exact limit theorems for restricted integer partitions

For a set of positive integers A, let pA(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy–Ramanujan formula for p(n) by showing that A has density α if and only if log⁡pA(n)∼log⁡p(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if log⁡pA(n)/log⁡p(αn) has lower limit 1. We answer this question negatively by constructing, for every α>0, a set of integers A of lower density α, satisfyinglim infn→∞log⁡pA(n)log⁡p(αn)≥(6π−oα(1))log⁡(1/α). We further show that the above bound is best possible (up to the oα(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

Partitions associated to class groups of imaginary quadratic number fields

We investigate properties of attainable partitions of integers, where a partition $$(n_1,n_2, \dots , n_r)$$ of n is attainable if $$\sum (3-2i)n_i\ge 0$$ . Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions

This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\mathcal{C}_{\mu}$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map that builds the tail part of each chain $\mathcal{C}_{\mu}$. Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain $\mathcal{C}_{\mu}$ for a (generalized) flagpole partition $\mu$, assuming that the chains indexed by certain specific smaller partitions (depending on $\mu$) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.